How to Find the Side of a Triangle Using the Angles

To calculate an unknown side of a triangle, you need only know the size of the angle opposite to it and one other length and angle. According to the trigonometric law of sines, a single ratio links each side's length with the sine of its opposing angle. Note, however, that you can't calculate a length if you know no lengths at all. A set of three angles corresponds with an infinite number of triangles of varying sizes.

Find the sine of the angle opposite to a known length. If this angle measures, for instance, 50 degrees, determine that sin(50) = 0.766.

Divide this answer by the known length. For this example, imagine that the length measures 5 inches, so 0.766 / 5 = 0.1532.

Find the sine of the angle opposite to the unknown side. If this angle measures, for instance, 85 degrees -- sin(85) = 0.996.

Divide this answer by the answer to Step 2 -- 0.996 / 0.1532 = 6.5 inches. This is the length of the unknown side.